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# Communications in Mathematical Sciences

## Volume 2 (2004)

### Number 1

### Hydrodynamic limits of Vlasov-Fokker-Planck equation for granular media

Pages: 121 – 136

DOI: http://dx.doi.org/10.4310/CMS.2004.v2.n1.a7

#### Authors

#### Abstract

This paper, which is a sequel to Benedetto-Caglioti-Golse-Pulvirenti, *Comput. Math. Appl.* **38** (1999), p. 121-131, considers as a starting point a mean-field equation for the dynamics of a gas of particles interacting via dissipative binary collisions. More precisely, we are concerned with the case where these particles are immersed in a thermal bath modeled by a linear Fokker-Planck operator. Two different scalings are considered for the resulting equation. One concerns the case of a thermal bath at finite temperature and leads formally to a nonlinear diffusion equation. The other concerns the case of a thermal bath at infinite temperature and leads formally to an isentropic Navier-Stokes system. Both formal limits rest on the mathematical properties of the linearized mean-field operator which are established rigorously, and on a Hilbert or Chapman-Enskog expansion.

#### Keywords

Granular media; Vlasov-Fokker-Planck equation; hydrodynamic limits; HIlbert expansion; Chapman-Enskog expansion

#### 2010 Mathematics Subject Classification

76M45, 76T25, 82C40